The least number , which is a perfect square and is divisible by each of the numbers 16, 20 and 24 is

Options

A. 1600
B. 3600
C. 6400
D. 14400

Answer

To find the least number that is a perfect square and divisible by 16, 20, and 24, we will follow these steps:

### Step 1: Find the LCM of the numbers 16, 20, and 24.
To find the LCM, we first need to determine the prime factorization of each number:

- **16**: \( 16 = 2^4 \)
- **20**: \( 20 = 2^2 \times 5^1 \)
- **24**: \( 24 = 2^3 \times 3^1 \)

### Step 2: Identify the highest powers of all prime factors.
Now we will take the highest power of each prime factor from the factorizations:

- For \(2\): The highest power is \(2^4\) (from 16).
- For \(3\): The highest power is \(3^1\) (from 24).
- For \(5\): The highest power is \(5^1\) (from 20).

### Step 3: Calculate the LCM.
Now we can calculate the LCM using these highest powers:

\[ \text{LCM} = 2^4 \times 3^1 \times 5^1 \]

Calculating this gives:

\[ \text{LCM} = 16 \times 3 \times 5 \]

Calculating step-by-step:

- \(16 \times 3 = 48\)
- \(48 \times 5 = 240\)

So, the LCM of 16, 20, and 24 is \(240\).

### Step 4: Ensure the LCM is a perfect square.
Next, we need to check if \(240\) is a perfect square. The prime factorization of \(240\) is:

\[ 240 = 2^4 \times 3^1 \times 5^1 \]

For a number to be a perfect square, all the powers in its prime factorization must be even. Here, the powers of \(3\) and \(5\) are odd.

### Step 5: Adjust the LCM to make it a perfect square.
To make \(240\) a perfect square, we need to adjust the odd powers:

- For \(3^1\), we need one more \(3\) to make it \(3^2\).
- For \(5^1\), we need one more \(5\) to make it \(5^2\).

Thus, we multiply \(240\) by \(3\) and \(5\):

\[ \text{Required number} = 240 \times 3 \times 5 \]

Calculating this:

\[ 240 \times 3 = 720 \]
\[ 720 \times 5 = 3600 \]

### Conclusion:
The least number that is a perfect square and divisible by 16, 20, and 24 is **3600**.

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